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A Quiet Platonism

The Fibonacci Sequence and

Whitehead's Concept of Ingression

**Flower Petals**: Many flowers have a number of petals that corresponds to a Fibonacci number (e.g., 3, 5, 8).**Sunflower Seed Heads**: The spiral pattern of seeds in a sunflower follows Fibonacci numbers.**Pinecones**: The arrangement of scales on a pinecone typically forms spirals that are Fibonacci numbers.**Pineapple Segments**: The spirals on a pineapple are arranged in Fibonacci numbers.**Nautilus Shell**: The growth of the nautilus shell follows a logarithmic spiral related to the Fibonacci sequence.**Leaf Arrangement (Phyllotaxis)**: Leaves around a stem often follow a spiral pattern corresponding to Fibonacci numbers.**Branching Trees**: The branching of trees often follows Fibonacci numbers, optimizing space and light exposure.**Fruit and Vegetables**: Examples include the spirals on the surface of Romanesco broccoli and the cross-section of apples.**Animal Reproduction**: The breeding pattern of rabbits can be modeled using Fibonacci numbers.**Hurricanes**: The spiral pattern of hurricanes resembles the Fibonacci spiral.**Galaxies**: Spiral galaxies, such as the Milky Way, often follow a Fibonacci spiral pattern.**DNA Molecule**: The double helix of DNA follows the Fibonacci sequence in its structural dimensions.**Human Ear**: The cochlea of the human ear is a Fibonacci spiral.**Fingers and Toes**: The number of bones in your fingers and toes follows the Fibonacci sequence.**Honeybee Family Tree**: The family tree of honeybees follows Fibonacci numbers.**Starfish**: Starfish typically have 5 arms, a Fibonacci number.**Spider Webs**: Some spider webs are built using the Fibonacci sequence.**Conch Shells**: The spiral shape of conch shells follows a Fibonacci pattern.**Animal Horns**: The spiral growth of animal horns (e.g., rams) follows the Fibonacci sequence.**Cactus Spines**: The arrangement of spines on some cacti follows Fibonacci spirals.

Whitehead's Concept of Ingression

The Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, is renowned for its appearance in various natural phenomena, from the arrangement of leaves on a stem to the spirals of shells and the branching of trees. For the mathematically minded, this mathematical sequence reflects an underlying order and harmony in the natural world, where the progression of growth and form often follows patterns that are elegant and seemingly inevitable.

Alfred North Whitehead’s concept of ingression provides a philosophical lens through which we can understand the connection between such mathematical patterns and the concrete realities we observe in the world. In Whitehead’s process philosophy, ingression refers to the manner in which eternal objects—abstract potentials or forms—become actualized within the world of concrete entities. Eternal objects are akin to Platonic forms; they are the pure possibilities that shape and influence the becoming of actual entities, which are the fundamental units of reality in Whitehead's metaphysical system. The difference is that, for Whitehead, these pure possibilities are not norms such as Truth, Goodness, and Beauty, and they are not "more real" than the world. Still, as with Plato, they are timeless: that is, not in temporal flux.

We might consider the Fibonacci sequence as an example of an eternal object in Whitehead's framework. As an abstract mathematical pattern, the Fibonacci sequence represents a potential form that can ingress into various actual entities in the natural world. This ingression does not mean that the sequence is merely imposed upon reality; rather, it suggests that the sequence is a potential that is realized in the very process of natural growth and formation.

When we observe the Fibonacci sequence manifesting in the spiral of a nautilus shell or the arrangement of seeds in a sunflower, we are witnessing the ingression of this eternal object into the concrete world. The sequence, as a mathematical possibility, becomes actualized in the structure of these natural entities. The ingression of the Fibonacci sequence contributes to the aesthetic and functional qualities of these forms, guiding the processes of growth and development in a way that aligns with the deeper order of the universe.

Whitehead's notion of ingression involves two key moments: the conception of an eternal object by a concrescing subject and the actualization of that object within the subject as it becomes a concrete entity. In the case of the Fibonacci sequence, we can think of natural processes as the concrescing subjects that "conceive" of the sequence as a potential form. This conception is not a conscious act but rather an inherent tendency within the natural process to align with certain patterns or structures that are conducive to growth and stability.

As these natural processes unfold, the Fibonacci sequence is actualized in the concrete forms we observe. This actualization is the culmination of the process, where the abstract pattern is fully realized in the physical world. The ingression of the Fibonacci sequence thus illustrates how abstract potentials, like mathematical sequences, can guide the development of concrete entities, bringing forth order and harmony from the realm of pure possibility.

The Fibonacci sequence, as it appears in the natural world, exemplifies Whitehead's concept of ingression—the process by which eternal objects, such as mathematical patterns, become actualized in the concrete realities of the universe. Through the lens of Whitehead’s philosophy, we can appreciate how abstract possibilities like the Fibonacci sequence are not just mathematical curiosities but are deeply embedded in the fabric of reality. The ingression of these eternal objects into the world of actual entities reveals the interconnectedness of abstract forms and concrete existence, highlighting the dynamic interplay between potentiality and actuality in the ongoing process of becoming that characterizes the universe.

John Cobb

After Whitehead has listed “categories of existence,” he identifies two of them as having “a certain extreme finality.” We have discussed one of these, the actual entities, at some length. The other one is “eternal objects,” and we will now turn to them. To get into the right ballpark, we can begin by saying that mathematical forms and formulae are eternal objects and that qualities of all kinds are eternal objects. E=mc2 is an eternal object; so is a definite shade of yellow. These eternal objects are directly illustrated in our world—in quite different ways. Anything that can be abstracted from experience and then can recur is an eternal object. There are also eternal objects that have never been actualized and never will be. A seven-dimensional space, also, is an eternal object, in that it can be thought about by mathematicians.

Whitehead himself gives the following equivalent terms: “pure potentials for the specific determination of fact,” and “forms of definiteness.” Occasionally he uses the term “abstract possibility,” and students have often made the contrast between eternal objects as pure possibilities and actual entities as possessing full actuality. However, Whitehead generally associates possibility with something that could actually occur. It is better to stay closer to his language. Eternal objects are pure potentials, and that means forms that could in principle characterize something actual, but that are in their nature indifferent to whether they do, or ever will, characterize anything actual. It is well to ask why Whitehead invented the term “eternal object” instead of sticking with the more familiar language of potentials and forms. First, “objects” establishes their status as depending on subjects. Objects exist only for subjects. They can be felt; they cannot feel. In themselves they cannot act. They are, indeed, passive.

For Whitehead “eternal” means nothing more than nontemporal. That is another way of saying that “eternal” objects have no actuality at all in themselves. They do not come into being and they do not pass away. They are related to every temporal moment in the same way, so far as their own nature is concerned.

Cobb Jr, John B. Whitehead Word Book: A Glossary with Alphabetical Index to Technical Terms in Process and Reality (Toward Ecological Civilization Book 8) . Process Century Press. Kindle Edition.

The Fibonacci Sequence:

A Scholarly Discussion

Melvyn Bragg and guests discuss the Fibonacci Sequence. Named after a 13th century Italian Mathematician, Leonardo of Pisa who was known as Fibonacci, each number in the sequence is created by adding the previous two together. It starts 1 1 2 3 5 8 13 21 and goes on forever. It may sound like a piece of mathematical arcania but in the 19th century it began to crop up time and again among the structures of the natural world, from the spirals on a pinecone to the petals on a sunflower.The Fibonacci sequence is also the mathematical first cousin of the Golden Ratio – a number that has haunted human culture for thousands of years. For some, the Golden ratio is the essence of beauty found in the proportions of the Parthenon and the paintings of Leonardo Da Vinci. With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Jackie Stedall, Junior Research Fellow in History of Mathematics at Queen’s College, Oxford; Ron Knott, Visiting Fellow in the Department of Mathematics at the University of Surrey

Interesting and Important?

1. **Mathematical Properties:**

2.**Appearance in Nature:**

3.**Applications in Science and Engineering:**

4.**Aesthetic and Artistic Significance:**

5.**Financial Markets:**

6.**Philosophical and Symbolic Value:**

7.**Historical Importance:**

The Fibonacci sequence captivates people due to its mathematical elegance, its pervasive presence in nature, its practical applications across various fields, and its deep connections to aesthetics and philosophy. It serves as a bridge between the abstract world of numbers and the tangible world we experience.

**Simple Definition, Complex Behavior**: The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. Despite its simple definition, the sequence exhibits intricate patterns and connections with various branches of mathematics.**Golden Ratio**: As the Fibonacci numbers increase, the ratio between consecutive Fibonacci numbers (e.g., Fn+1Fn\frac{F_{n+1}}{F_n}FnFn+1) approaches the Golden Ratio, approximately 1.618. The Golden Ratio is deeply embedded in many natural and human-made structures, often associated with aesthetic beauty.**Binet's Formula**: There is an explicit formula to find the nnnth Fibonacci number without generating the sequence, showing the deep connection between algebra and the sequence.

2.

**Phyllotaxis**: The Fibonacci sequence appears in the arrangement of leaves, seeds, and petals in plants. For example, the number of petals in many flowers is often a Fibonacci number, and the spirals on pinecones, pineapples, and sunflowers correspond to Fibonacci numbers.**Animal Behavior**: The breeding patterns of rabbits, which Fibonacci originally described in his 1202 book*Liber Abaci*, can be modeled using Fibonacci numbers, although the real-world scenario is more complex.

3.

**Computer Science**: The Fibonacci sequence is used in algorithm analysis, especially in recursive algorithms, dynamic programming, and data structure design (e.g., Fibonacci heaps).**Optimization**: The Fibonacci method is used in optimization problems, particularly in the Fibonacci search technique, to efficiently find the minimum of unimodal functions.**Fractals and Chaos Theory**: Fibonacci numbers often appear in the study of fractals, which are structures that repeat at every scale. They are also connected to the study of dynamical systems and chaos theory.

4.

**Architecture and Art**: The Golden Ratio, closely related to the Fibonacci sequence, is often associated with aesthetically pleasing proportions. Many artists and architects have employed these proportions in their work, from ancient Greek temples to Renaissance paintings and modern design.**Music**: The Fibonacci sequence has been used in the structure of musical compositions. Some composers have used Fibonacci numbers to determine the lengths of sections or even the frequencies of notes.

5.

**Technical Analysis**: Traders and analysts in financial markets sometimes use Fibonacci retracement levels to predict potential price movements. These levels are based on Fibonacci ratios (23.6%, 38.2%, 61.8%) and are believed to be significant in determining support and resistance levels in the markets.

6.

**Universal Pattern**: The ubiquity of the Fibonacci sequence in natural patterns leads some to see it as a universal rule or a symbol of the inherent order in the universe.**Mathematical Beauty**: Many find the sequence beautiful due to its simplicity, surprising appearances in diverse contexts, and the elegant relationships it reveals.

7.

**Introduction of the Hindu-Arabic Number System**: Fibonacci's*Liber Abaci*was one of the first books to introduce the Hindu-Arabic numeral system to Europe, promoting the use of the zero and positional notation that is now standard worldwide.**Influence on Mathematics**: The sequence has inspired a broad range of mathematical studies, from number theory to combinatorics and beyond.

The Fibonacci sequence captivates people due to its mathematical elegance, its pervasive presence in nature, its practical applications across various fields, and its deep connections to aesthetics and philosophy. It serves as a bridge between the abstract world of numbers and the tangible world we experience.

The Spiritual Implications:

A Quiet Platonism

My friend Lisa, a mathematician by profession and self-described "numbers lover," has always found solace in the precision and elegance of mathematics. She isn't sure if she believes in God, but she does believe in the power and truth of mathematics. To her, mathematical patterns aren't just symbols on a page; they are the building blocks of reality, the hidden language that shapes the world around us.

On Sunday, she invites me over to her home, eager to share her latest collection of what she calls her "Fibonacci scrapbook." It’s a scrapbook she keeps to continually add photos of where the Fibonacci sequence—a mathematical pattern where each number is the sum of the two preceding ones, starting with 0 and 1—is at work.

As she spreads the photos out on the table, I am struck by the sheer diversity of subjects—each one an example of the Fibonacci sequence in nature. "Look at this," she says, pointing to a close-up of a sunflower head. The spiral patterns of its seeds follow the Fibonacci sequence perfectly, each number in the series adding up to create the next, the pattern radiating outward in a beautiful display of mathematical order.

"See how it’s everywhere?" Janet continues, showing me another photo, this time of a pine cone. The scales of the cone spiral in a way that echoes the same sequence. "It’s in the branches of trees, in the shells of snails, even in the way galaxies are formed. It’s like the universe is written in this language, and we’re just beginning to understand it."

As we sit there, surrounded by images from her Fibonacci scrapbook, I can’t help but marvel at Janet’s deep connection to these patterns. In many ways, I think of her as a kind of Platonist—an outlook so unfashionable in modern times. While the world rushes forward with empirical science and technological advancements, Janet finds truth in the abstract, in the timeless forms that underlie the chaos of everyday life. For her, the Fibonacci sequence isn’t just a mathematical curiosity—it’s a form of truth, something she can hold onto in a world full of uncertainties.

She knows that I am a theologian. She wants to suggest that she, in her way, is a theologian of sorts, too. Though she might question the existence of a higher power, her belief in mathematics is inspiring. In the patterns and sequences she sees all around her, Janet finds a sense of order and beauty that speaks to her on a profound level, offering a glimpse of something greater, even if it isn’t divine. In a world that often overlooks the abstract, Janet’s quiet Platonism feels like a breath of fresh air, a reminder that some truths lie beyond the reach of sense perception, waiting to be discovered in the elegant patterns of the natural world. Strangely enough, Janet's quiet Platonism does not take her further from the world, it brings her closer to the world. I think the philosopher who so influences me and other process philosophers and theologians, Alfred North Whitehead, would like her.

- Jay McDaniel

On Sunday, she invites me over to her home, eager to share her latest collection of what she calls her "Fibonacci scrapbook." It’s a scrapbook she keeps to continually add photos of where the Fibonacci sequence—a mathematical pattern where each number is the sum of the two preceding ones, starting with 0 and 1—is at work.

As she spreads the photos out on the table, I am struck by the sheer diversity of subjects—each one an example of the Fibonacci sequence in nature. "Look at this," she says, pointing to a close-up of a sunflower head. The spiral patterns of its seeds follow the Fibonacci sequence perfectly, each number in the series adding up to create the next, the pattern radiating outward in a beautiful display of mathematical order.

"See how it’s everywhere?" Janet continues, showing me another photo, this time of a pine cone. The scales of the cone spiral in a way that echoes the same sequence. "It’s in the branches of trees, in the shells of snails, even in the way galaxies are formed. It’s like the universe is written in this language, and we’re just beginning to understand it."

As we sit there, surrounded by images from her Fibonacci scrapbook, I can’t help but marvel at Janet’s deep connection to these patterns. In many ways, I think of her as a kind of Platonist—an outlook so unfashionable in modern times. While the world rushes forward with empirical science and technological advancements, Janet finds truth in the abstract, in the timeless forms that underlie the chaos of everyday life. For her, the Fibonacci sequence isn’t just a mathematical curiosity—it’s a form of truth, something she can hold onto in a world full of uncertainties.

She knows that I am a theologian. She wants to suggest that she, in her way, is a theologian of sorts, too. Though she might question the existence of a higher power, her belief in mathematics is inspiring. In the patterns and sequences she sees all around her, Janet finds a sense of order and beauty that speaks to her on a profound level, offering a glimpse of something greater, even if it isn’t divine. In a world that often overlooks the abstract, Janet’s quiet Platonism feels like a breath of fresh air, a reminder that some truths lie beyond the reach of sense perception, waiting to be discovered in the elegant patterns of the natural world. Strangely enough, Janet's quiet Platonism does not take her further from the world, it brings her closer to the world. I think the philosopher who so influences me and other process philosophers and theologians, Alfred North Whitehead, would like her.

- Jay McDaniel

Fibonacci Numbers

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